Create a menu item using wp_get_nav_menu_items. This simply adds the item to the end. The cool thing it appears in the menus editor too! It would be nice to add ability to sort in the menu and remove the item using the menu editor.

wp_get_nav_menu_items-filter

// from http://www.daggerhart.com/dynamically-add-item-to-wordpress-menus/

add_filter( 'wp_get_nav_menu_items', 'custom_nav_menu_items', 20, 2 );

function custom_nav_menu_items( $items, $menu ){
    // only add item to a specific menu
    if ( $menu->slug == 'secondary-navigation' ){

        // only add profile link if user is logged in
        if ( get_current_user_id() ){
            $items[] = _custom_nav_menu_item( 'Logout', wp_logout_url( '/' ), 3 ); //log user out and drop them on the home page
        }
    }

    return $items;
}

/**
 * Simple helper function for make menu item objects
 * 
 * @param $title      - menu item title
 * @param $url        - menu item url
 * @param $order      - where the item should appear in the menu
 * @param int $parent - the item's parent item
 * @return \stdClass
 */ 
function _custom_nav_menu_item( $title, $url, $order, $parent = 0 ){
  $item = new stdClass();
  $item->ID = 1000000 + $order;
  $item->title = $title;
  $item->url = $url;
  $item->menu_order = $order;
  $item->menu_parent_item = $parent;
  $item->type = '';
  $item->object = '';
  $item->object_id = '';
  $item->db_id = '';
  $item->classes = array();
  
  return $item;
}

I think line 33 should be

$item->menu_item_parent = $parent;